What would have made you feel better? My guess is that you would have been happy if the other person wasn't upset with what you said and didn't disagree with you, right? Do you think if you were able to explain yourself, then the other person would have understood you and not disagreed with you?

It's likely that would have not been the case. There is a very good chance that they still would have disagreed with you even after elaboration. And you know what? It's not your fault.

You will have a large set of views about different facets of life. Even if they were all the most sane, rational views, many people will disagree with them. (And in different combinations. You may have Andy agree with you on X, Y, and Z but disagree on Q, R, and S. Brad may think you're right with X, R, and S but disagree on the rest.)

It is inevitable. So, what is someone to do?

First, is it something that matters? If it's something like an opinion of which celebrity has the best smile, remind yourself it doesn't matter.

But if it's something that does matter? Make sure you educate yourself. Accept evidence to the contrary to what you believe (from reliable sources). Keep an open mind. Accept input. Be aware of your own bias. If you need to update your own viewpoint because you found out you were wrong... Then do it! Yes, it sucks that you were wrong. But it's better to have been wrong then correct yourself than to stay wrong. This is important... If you're wrong, act the way you would want the other person to act if they were wrong. (You will make the world a better place doing this.)

Now, does this person still not agree with you (and you updated your own viewpoint based on facts)? Can you change their mind? Probably not. Is this a failing on your part? No. You can't control other people, just like they can't control you. But you can control yourself.

Being told we're wrong sucks. But if you do not have sufficient evidence that you are wrong, then you should be confident in what you think. Instead of framing this scenario as "this person disagrees with me," frame it as "I disagree with this person." With time and practice, you will more easily move to "I disagree with this person, and I am ok with it."

One last note. If it's something that is very important to you, make sure to do what you can to make the change you want to see in the world. If you were very concerned about pollution for instance, do things like trash tag, buying less stuff, and advocating for your cause. This specific person may not help, but you can still live your life as a reflection of your own values to the best of your ability and maybe even collaborate with others as well.

I hope this helps.